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248                      ANALYTICAL MECHANICS
center of mass plus the kinetic energy of its parts due to their motion relative to the center of mass. Collision does not affect the motion of the center of mass of the system formed of the colliding bodies, because the forces which arise during the collision are internal forces. Therefore that part of its kinetic energy which is due to the motion of its center of mass does not suffer any loss. The loss occurs in that part of the energy which is due to the motion of the parts of the system with respect to the center of mass. Referring all the velocities to the center of mass and denoting the loss of kinetic energy by Th we have
Tl = (i m^ + J m2^22) - (| w^i'2 + 1 m2?;2'2),
where vi and v% are the velocities just before and v\ and vj the velocities just after the collision.
We can eliminate vą and v% from this expression for Tl by means of the principle of the conservation of momentum and the definition of e. According to the former
rriiVi + m2v2 = rriiVi + m^r and by (X')
vi - v2' = -e (vi - tfe).
Eliminating %' between the last two equations we have
The following changes in the expression of Tl are effected by means of the last three equations.
Tl = | mfa* - v/2) + i m2 fe2 - *;2'2) = | ml (vi - vfl
= | ml ^ - vflfa - %)(! - e)
When the colliding bodies are perfectly elastic then e = 1